Appendix B — Mathematical Reference

A quick-reference summary of the mathematics used throughout this book.

B.1 Linear Algebra

Key Operations

Eigendecomposition

For symmetric $A$: $$A = Q \Lambda Q^\top$$ where $Q$ is orthogonal (columns are eigenvectors) and $\Lambda = \text{diag}(\lambda_1, \ldots, \lambda_n)$.

Solving Linear Systems

$Ax = b$ solved by LU decomposition: $O(n^3)$ complexity.

B.2 Calculus and Optimisation

Taylor Series (second order)

$$f(x + \delta) \approx f(x) + f'(x),\delta + \tfrac{1}{2} f''(x),\delta^2$$

Integration by Parts

$$\int_a^b u,dv = [uv]_a^b - \int_a^b v,du$$

Gradient Descent

$$\theta_{k+1} = \theta_k - \eta \nabla_\theta \mathcal{L}(\theta_k)$$

Convergence guaranteed for $L$-smooth convex $\mathcal{L}$ with $\eta < 1/L$.

B.3 Probability and Statistics

Normal Distribution

$$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}, \qquad \Phi(x) = \int_{-\infty}^x \phi(t),dt$$

Useful identities:

• $\Phi(-x) = 1 - \Phi(x)$
• $E[e^{\sigma Z}] = e^{\sigma^2/2}$ for $Z \sim N(0,1)$

Log-Normal Distribution

If $X = e^{\mu + \sigma Z}$, $Z \sim N(0,1)$: $$E[X] = e^{\mu + \sigma^2/2}, \qquad \text{Var}(X) = e^{2\mu+\sigma^2}(e^{\sigma^2}-1)$$

Moment Generating Function

$$M_X(t) = E[e^{tX}]$$

For $X \sim N(\mu, \sigma^2)$: $M_X(t) = e^{\mu t + \sigma^2 t^2/2}$.

B.4 Stochastic Calculus

Brownian Motion Properties

• $W_0 = 0$
• $W_t - W_s \sim N(0, t-s)$ for $t > s$
• Independent increments
• Continuous paths (almost surely)

Itô's Lemma

For $f(t, W_t)$ twice differentiable: $$df = \frac{\partial f}{\partial t},dt + \frac{\partial f}{\partial x},dW_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2},dt$$

Geometric Brownian Motion

$$dS_t = \mu S_t,dt + \sigma S_t,dW_t$$ $$S_t = S_0 \exp!\left[\left(\mu - \tfrac{\sigma^2}{2}\right)t + \sigma W_t\right]$$

Girsanov Theorem

Under measure $\mathbb{Q}$ defined by: $$\frac{d\mathbb{Q}}{d\mathbb{P}} = \exp!\left(-\int_0^T \theta_t,dW_t - \tfrac{1}{2}\int_0^T \theta_t^2,dt\right)$$ the process $\tilde{W}_t = W_t + \int_0^t \theta_s,ds$ is a $\mathbb{Q}$-Brownian motion.

B.5 Black-Scholes Formula Quick Reference

$$C = S,\Phi(d_1) - K e^{-rT}\Phi(d_2)$$ $$P = K e^{-rT}\Phi(-d_2) - S,\Phi(-d_1)$$

$$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$$

Greeks

B.6 Standard Normal Table (selected values)

B.7 Key Financial Formulas

Bond Duration and Convexity

$$D = \frac{1}{P}\sum_{i=1}^n \frac{t_i \cdot C_i}{(1+y)^{t_i}}, \qquad \text{Cx} = \frac{1}{P}\sum_{i=1}^n \frac{t_i(t_i+1)\cdot C_i}{(1+y)^{t_i+2}}$$

Nelson-Siegel Yield Curve

$$y(\tau) = \beta_0 + (\beta_1 + \beta_2)\frac{1 - e^{-\tau/\lambda}}{\tau/\lambda} - \beta_2 e^{-\tau/\lambda}$$

Vasicek Short Rate

$$dr_t = \kappa(\theta - r_t),dt + \sigma,dW_t$$

Zero-coupon bond: $P(t,T) = A(t,T)e^{-B(t,T)r_t}$ where: $$B(t,T) = \frac{1 - e^{-\kappa(T-t)}}{\kappa}$$ $$\ln A(t,T) = \left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(t,T) - (T-t)) - \frac{\sigma^2}{4\kappa}B(t,T)^2$$